arX

iv:h

ep-t

h/03

0104

8v2

17

Jan

2003

hep-th/0301048

Deconstruction, Lattice Supersymmetry,

Anomalies and Branes

Joel Giedta, Erich Poppitza, and Moshe Rozalib

aDepartment of Physics, University of Toronto

Toronto, ON, M5S 1A7, Canada

bDepartment of Physics and Astronomy, University of British Columbia

Vancouver, BC, V6T 1Z1, Canada

giedt@physics.utoronto.ca, poppitz@physics.utoronto.ca,

rozali@physics.ubc.ca

Abstract

We study the realization of anomalous Ward identities in deconstructed (latticized)

supersymmetric theories. In a deconstructed four-dimensional theory with N = 2

supersymmetry, we show that the chiral symmetries only appear in the infrared and

that the anomaly is reproduced in the usual framework of lattice perturbation theory

with Wilson fermions. We then realize the theory on the world-volume of fractional

D-branes on an orbifold. In this brane realization, we show how deconstructed theory

anomalies can be computed via classical supergravity. Our methods and observations

are more generally applicable to deconstructed/latticized supersymmetric theories in

various dimensions.

1 Introduction and Summary

Deconstruction was originally proposed as a gauge-invariant regularization and ultraviolet

completion of five- and six-dimensional gauge theories in terms of a four-dimensional theory

with a particular product gauge group, a “moose” theory [1]. (Theories with a product of

N gauge groups and matter fields linking the groups were considered earlier in [2] and [3],

albeit with a very different motivation and interpretation.)

These “moose,” or “quiver,” product-group theories appear naturally in string theory as

world-volume theories on branes at orbifold singularities [4]. The relation of deconstruction

to string theory has led to some interesting proposals. One is the idea that a four dimensional

“moose” theory may provide, in an appropriate limit, a definition of the ill-understood (0, 2)

supersymmetric six-dimensional theory [5].

Another proposal is based on the observation that the brane realization of deconstruction

can be used to give a definition of three- and four-dimensional supersymmetric gauge theories

in terms of one- or zero-dimensional product-group theories. These one- (zero-) dimensional

theories can provide a spatial [6] (or Euclidean [7]) lattice formulation of supersymmetric

theories; see also [8]. The lattice theory explicitly preserves part of the supersymmetry of

the continuum theory. The explicit supersymmetry, along with other symmetries of the

deconstructed theory forbids relevant operators that break the continuum supersymmetry,

and ensures that the enhanced continuum supersymmetry is achieved without fine tuning

[6].

This development leads to the hope that deconstruction may ultimately be useful to

study strong-coupling supersymmetric gauge dynamics in three or four dimensions via nu-

merical simulations; for a review of the progress in this direction, see [9]. This is important,

because many interesting aspects of supersymmetric gauge dynamics, relevant for various

applications, are not accessible by the tools—holomorphy and symmetries—normally used

to obtain exact results.

An important tool in the study of nonperturbative gauge dynamics is the understanding

of chiral symmetries and anomalies (both gauge and ’t Hooft). In this paper, we investi-

gate the chiral symmetries and their anomalies in the deconstructed (lattice) formulation

of supersymmetric theories. We note that some related issues—the parity anomaly and

the Chern-Simons term—in the deconstruction of five-dimensional theories have been ad-

dressed before [10, 11]; see also [12]. Here we study the deconstruction of four-dimensional

1

theories, with continuous global chiral symmetries with gauge or ’t Hooft anomalies. The

deconstructed supersymmetric theories we study are vector-like. Applying deconstruction

to chiral gauge theories can be done in principle, but is expected to face the usual problems

of putting chiral fermions on the lattice. We hope to return to this issue in the near future.

We consider a simple example: the four-dimensional N = 2 supersymmetric pure Yang-

Mills theory on a one-dimensional spatial lattice, i.e. its deconstruction in terms of a three-

dimensional product-group theory. This setup is relatively simple, allowing us to look at

chiral symmetries and anomalies from different points of view, while keeping the expressions

reasonably tractable; yet, it contains features generic enough to be shared by other decon-

structed/lattice supersymmetric theories (e.g. 4d N = 4, or 3d N = 4, SYM on a spatial

lattice [6]; the deconstruction of higher-dimensional supersymmetric theories, etc.). In fact,

all our findings can be generalized to these examples.

Below, we give a summary of our results and point to the relevant sections of the paper:

1. We find that the chiral symmetry of the theory is not explicit at finite lattice spacing

and is broken by a Wilson term, with coefficient fixed by the (super-)symmetries of

the deconstructed theory. Thus the chiral symmetry appears only as an accidental

symmetry in the continuum limit. The lattice action is worked out in section 2.1, and

the equivalence to the standard Wilson fermion formulation is shown in section 2.2.

2. The anomaly of the chiral U(1)R symmetry can be understood using lattice perturba-

tion theory, as in the classic work of Karsten and Smit [13]. A point worth noticing

is that the large value of the Wilson term coefficient precludes the interpretation of

the anomaly as due to heavy doublers acting as 4d Pauli-Villars regulators in the con-

tinuum limit, yet the correct anomalous Ward identity results. This is discussed in

section 2.2.

3. A “modern” view on the chiral anomaly in the deconstructed theory results from con-

sidering its fractional brane realization. This brane realization is explained in sections

3.1, 3.2. In section 3.1, we also review how the U(1)R anomaly in the continuum

Seiberg-Witten theory is understood from the corresponding classical supergravity.

4. We show, in section 3.3, how one can compute the anomaly in the lattice version of the

theory using fractional branes. We also find the expression for the running coupling

and discuss the continuum limit. In section 3.4, we give a geometric interpretation of

the R-symmetry violating Wilson term.

2

To summarize, we have shown that even though the chiral symmetries are explicitly broken in

the deconstructed/latticized supersymmetric theories, the correct anomalous Ward identities

are obtained in the continuum limit. We show this from two different points of view: lattice

perturbation theory and the brane realization of deconstruction. Our observations are more

generally valid than the model we considered: for example, fractional branes can be used to

study the gauge coupling running and the Chern-Simons coefficient in deconstructed N = 1

5d theories.

We do not yet have a detailed understanding of the anomaly in the deconstructed theory

as charge nonconservation in a nontrivial gauge field background. This question is currently

under investigation; it requires a better understanding of the map between nonperturbative

effects in the deconstructed and continuum theories.

2 Global Chiral Symmetries and Anomalies in the De-

constructed Seiberg-Witten Theory

In this section, we study the global chiral symmetries of the classical action of the

deconstructed N = 2 4d Yang-Mills theory. We show that the global chiral SU(2)R ×U(1)R

symmetry of the continuum theory is explicitly broken by the lattice action. The anomaly of

the U(1)R symmetry is reproduced, in the continuum limit, by lattice perturbation theory,

similar to [13].

2.1 The Deconstructed Theory Action and Symmetries

We begin by writing down the action of the deconstructed theory, starting from the super-

space expression. The theory whose lattice action we want to explicitly study is the de-

construction of a 4d N = 2 Yang-Mills theory with gauge group SU(k). Its deconstruction

is the dimensional reduction, from 4d to 3d, of the theory considered in [14]. It is conve-

nient to retain four-dimensional superfield notation throughout, while simply “forgetting”

the dependence on the coordinate x3.

The deconstructed theory is a SU(k)N quiver gauge theory1 with N = 2 3d supersym-

1For simplicity, we ignore the U(1) factors present in the lattice models of [6], as well as in the D-brane

construction, which have a U(k)N gauge group. We only note that the strength of the U(1) D-term potential

might be used to stabilize the relative fluctuations of the Qi vevs (2.3), i.e. the difference between lattice

spacings on neighboring links (but not of the overall zero mode; this issue is not relevant for our 3d to 4d

3

metry (four real supercharges). The gauge fields are in vector supermultiplets Vi, while

the chiral matter superfields Qi transform as a fundamental under the i-th gauge group

and antifundamental under the i + 1-th gauge group. In other words Qi → eΛiQie−Λi+1 ,

eVi → e−Λ†i eVie−Λi under gauge transformations. The N + 1-th gauge group is identified

with the first, so that a closed circular “moose” is obtained. The superspace lagrangian is

described by the tree-level Kahler potential:

K =1

g23

N∑

i=1

tr Q†i e

Vi Qi e−Vi+1 . (2.1)

The F -term part of the tree-level lagrangian2 arises from the gauge kinetic terms:

W =1

2g23

N∑

i=1

tr W α(Vi) Wα(Vi) . (2.2)

The superpotential is forbidden by the U(1)R symmetry of the theory, part of the U(1)N ×U(1)R × ZN global symmetry of the model (here U(1)i is the global symmetry under which

the Qi have unit charge and ZN is the “rotational” symmetry of the moose; the theory also

preserves 3d parity). We keep the canonical dimensions of all fields appropriate to 4d, so

g23 has dimension of mass. We use generators in the fundamental, so that tr T aT b = δab/2

and [T a, T b] = ifabcT c. Further, we also need gabc = tr(T aT bT c) = 14ifabc + 1

2dabc, with

dabc = tr T a{T b, T c}.Below, we work in Wess-Zumino gauge. We let m = 0, 1, 2 and µ = 0, 1, 2, 3. The A3

component of the 4d vector boson is the real adjoint scalar in the 3d N = 2 vector multiplet.

Even though in 3d we can use the charge conjugation matrix σ3αα to convert dotted into

undotted indices and write the entire lagrangian in terms of one type of indices, we will keep

the dotted and undotted spinor indices as this simplifies the consideration of the 4d limit.

As was shown in [14] by considering the mass spectrum, the theory (2.1), (2.2) describes,

along the Higgs branch 〈Qi〉 = v and in the large-N limit, a supersymmetric field theory with

gauge group SU(k) in one higher dimension and with twice the amount of supersymmetry,

in our case—an N = 2 4d supersymmetric theory.

Our aim here is to study the interactions and the symmetries of the theory. To describe

the relevant (i.e. the ones charged under SU(k)N , keeping the vevs frozen, see footnote 1)

fluctuations of the link fields Qi, we define:

Qi = v (1 + qai Ta) . (2.3)

deconstruction, so we do not address it here).2In the superrenormalizable N = 2 3d supersymmetric theory (2.1), (2.2), there are no divergent correc-

tions to the tree level lagrangian. Finite loop effects are discussed in section 3.3.

4

It is straightforward to isolate the fermion bilinear terms corresponding to (2.1) and (2.2):

Lferm. =|v|2g23

N∑

j=1

[

− i

2ψaj σ

m∂mψaj +

i

2√2ψaj (λ

aj − λaj+1)−

i

2√2ψaj (λ

aj − λaj+1)

−1

2gabc ψa

j σµψb

j Aj+1,cµ +

1

2gabc ψa

j σµψc

j Aj,bµ +

+i√2gabc

(

qbj λcj+1ψ

aj − q∗,aj λcj+1ψ

bj

)

− i√2gabc

(

qcj λbjψ

aj − q∗,aj λbjψ

cj

)

]

+1

g23

N∑

j=1

[

−iλaj σm∂mλaj +

i

2fabcλaj σ

µλcjAj,bµ

]

. (2.4)

The expression (2.4) will be useful in section 2.2, when comparing to the Wilson fermion

action. However, we will first consider the fermion lagrangian (2.4) in momentum space

along the latticized direction. Diagonalizing the kinetic terms helps to reveal the 4d nature

of the theory and the enhanced supersymmetry in the continuum limit, as well as to elucidate

the finite-N symmetries of the fermion interactions.

To write down the Fourier space action, we will proceed in two steps, skipping the

intermediate results, but explicitly giving the relevant field redefinitions. First, we perform

the following Fourier transformations in (2.4), assuming odd N for definiteness:

λj =1√N

N−12∑

k=−N−12

ωjk− k

2N λk, ψj =

√2√

N |v|

N−12∑

k=−N−12

ω−jkN ψk ,

Ak,aµ =

1√N

N∑

j=1

Aj,aµ ω

jk− k2

N , qak =|v|√2N

N∑

j=1

qaj ωjkN ,

q∗ak =|v|√2N

N∑

j=1

q∗aj ω−jkN = (qak)

∗, (2.5)

where ωN = e2πiN , and the complex conjugate transformation for the dotted fields. Note also

the reality condition A−k = (Ak)∗. It is easy to verify that terms proportional to dabc vanish

in the continuum limit. We will thus (for simplicity only) further ignore them even for finite

N , concentrating, e.g., on the SP (n) case. Finally, note that the lagrangian is invariant under

3d parity, with ψ(−x2) = iσ3σ2ψ(x), λ(−x2) = −iσ3σ2λ(x2), A2(3)(−x2) = −A2(3)(x2), and

A0(1)(−x2) = A0(1)(x2). Invariance of the mass term requires opposite intrinsic parities of

the gauginos and bifundamental fermions.

The second step in revealing the four-dimensional Lorentz invariant nature of the con-

tinuum limit spectrum is to perform yet another field redefinition:

ψkα =1√2

(

σ3ααχ

αk + iηkα

)

, ψαk =

1√2

(

−σ3ααχkα − iηαk)

,

5

λkα =1√2

(

χkα + iσ3ααη

αk

)

, λαk =1√2

(

χαk + iσ3ααηkα

)

. (2.6)

and write the final result for the fermion bilinear lagrangian (2.4) in Fourier space as:

Lferm. =1

g23

N−12∑

k=−N−12

[

− i(

ηakσm∂mη

ak + χa

−kσm∂mχ

a−k

)

+ |v| sin πkN

(

ηak σ3ηak + χa

−kσ3χa

−k

)

]

+1√Ng23

N−12∑

k,k′=−N−12

[

− i

2fabcBc,k,k′

µ

(

ηakσµηbk′ + χa

−kσµχb

−k′

)

+1

2fabc

(

ηakχbk′ φ

∗ck,k′ + ηakχ

bk′ φ

ck,k′

)

]

. (2.7)

In the continuum limit (|v| → ∞) all modes except those with k ≪ N decouple and the

lagrangian (2.7) becomes that of the 4d N = 2 supersymmetric Yang-Mills theory. We have

defined the Fourier components (along the latticized direction) of the 4d gauge field Bµ and

the 4d complex adjoint scalar φ as follows:

Bc,k,k′

m = Ac,k−k′

m cos2(

π(k − k′)

2N

)

,

Bc,k,k′

3 = − i√2

(

cosπk

Nq∗,ck′−k − cos

πk′

Nqck−k′

)

,

φck,k′ = Ac,k−k′

3 cos2(

π(k − k′)

2N

)

− i√2

(

cosπk

Nq∗ck′−k + cos

πk′

Nqck−k′

)

. (2.8)

Note that the third component of the 4d gauge field, B3, is the imaginary part of the link

fields’ fluctuations, qi of (2.3), while the scalar in the 3d vector multiplet, A3, combines

with the real part of the link fields to form the complex 4d adjoint scalar φ. Some further

comments are in order:

1. The fields ηk, η−k and χ−k, χk in (2.7) are the left and right moving components of

the two 4d adjoint fermions in the continuum limit, respectively. The free part of the

fermion action respects an U(2)N symmetry, with (ηak , χa−k)

T transforming as a doublet

of SU(2)k with unit charge under the corresponding U(1)k.

2. The fermion bilinear lagrangian Lferm. preserves, even for finite N , a diagonal SU(2)×U(1) ⊂ U(2)N symmetry, with k-independent action on the fermions as described

6

in the previous paragraph. Note that, even though the current associated with this

diagonal U(1) has the correct low-energy limit to represent (the 3d part of) a 4d U(1)R

current, it does not obey the correct anomalous Ward identity. As will be shown below,

there exists a current with the same R charges for the light states, but with different

R-charges for the heavy states, which gives the correct anomaly.

3. We should stress that the above diagonal SU(2)×U(1) symmetry of the fermion bilinear

terms is not a symmetry of the lagrangian and is broken by the bosonic terms—it is

evident from the definition (2.8) of the 4d scalar adjoint φ that one can not even

consistently define an U(1) action on φ for finite N (so that the kinetic terms of A3

and q are invariant). We thus conclude that the classical SU(2) × U(1) R-symmetry

of the 4d N = 2 theory is only exact in the continuum limit.

4. As usual in deconstruction, the mass spectrum can be read off from eqn. (2.7). For

k ≪ N , it approximates the Kaluza-Klein spectrum of a four-dimensional theory com-

pactified on a circle of radius:

R =N

π|v| . (2.9)

The value of the lattice spacing is a = 2/|v|.

The above analysis applies also to models of lattice supersymmetry with two or more latti-

cized dimensions, but the classical action and field redefinitions turn out to be a lot more

complicated and there is no point of giving them here—as we will see (see section 3.4), the

D-brane picture makes the finite-N violation of the chiral symmetries explicit.

2.2 Equivalence to Wilson Fermion Action and the Anomaly Cal-

culation

We now go back to the x-space lattice lagrangian (2.4). It has already been noted that

deconstruction of 5d theories gives rise to Wilson terms for the fermions3 [15]; this, in fact,

is a generic feature of deconstruction. Here, we show this using the deconstructed theory

action obtained in section 2.1. More generally, in section 3.4, we will see that the Wilson

term has a simple geometric interpretation in the brane realization of deconstruction.

Let us compare our lattice fermion action (2.4) to the one used in the work on anomalies

with Wilson fermions of Karsten and Smit (KS) [13]. We start with the usual Wilson fermion3One of us (E.P.) thanks Chris Hill for discussions.

7

action on a 4d lattice (see e.g. [13]). We convert to a 2-component notation, where the Dirac

spinor ψ of KS is4

ψ =

(

χα

ηα

)

, (2.10)

set the bare mass to zero, take the continuum limit in x0, x1 and x2, and obtain the Wilson

fermion lagrangian with one latticized dimension:

LKSferm. =

∑

j

{

− i [ηjσm∂mηj + χj σ

m∂mχj ]

− i

2a

[

ηjσ3(ηj+1 − ηj−1) + χj σ

3(χj+1 − χj−1)]

+r

2a[ηj(χj+1 + χj−1 − 2χj) + ηj(χj+1 + χj−1 − 2χj)]

}

. (2.11)

We compare this to the free part of (2.4):

Lfreeferm. =

1

g23

∑

j

{

−i[

|v|22ψj σ

m∂mψj + λj σm∂mλj

]

− i|v|2

2√2

[

ψj(λj+1 − λj)− ψj(λj+1 − λj)]

}

. (2.12)

Recall that the lattice spacing is a = 2/|v|. The field redefinitions relating the deconstructed

theory lagrangian (2.12) to the Wilson fermion lagrangian (2.11) are:5

|v|g3√2ψjα =

1√2(σ3

ααχαj − iηjα), (2.13)

− 1

g3λjα =

1√2(χjα − iσ3

ααηαj ).

The matching of the two lagrangians shows that r = 1 in the deconstructed theory. This

value of the Wilson term is imposed by the explicit supersymmetry and gauge invariance of

the deconstructed theory.

In the four-component basis of (2.10), the axial U(1) current is associated with ψ →e−iϕγ5ψ with γ5 = diag (−1, 1). Making this local we have in the 2-component notation:

ηj → eiϕjηj , χj → eiϕjχj . (2.14)

4ψ, η and χ here should not be confused with the spinors of the previous subsection.5Note that these are related to (2.6) by k → j and η → −η; the sign is important in the Wilson mass

term.

8

This axial transformation in our deconstructed theory is the anomalous U(1)R transformation

of the continuum 4d theory, combined with a diagonal generator of the anomaly free SU(2)R.

The variation of the lattice action (2.11), or, equivalently, (2.4) written in terms of the

new variables (2.13), under the axial U(1) is:6

δLKSferm.

δϕj

= −∂m [ηj σmηj + χj σ

mχj]

− 1

2a

[(

ηj σ3ηj+1 + χj σ

3χj+1 + h.c.)

−(

ηj−1σ3ηj + χj−1σ

3χj + h.c.)]

+r

2a[iηj(χj+1 + χj−1 − 2χj) + iχj(ηj+1 + ηj−1 − 2ηj) + h.c.] . (2.15)

In the continuum limit, the second term in brackets becomes:

− ∂3[

ησ3η + χσ3χ]

. (2.16)

Thus the anomalous Ward identity is given by:

〈∂µ [ησµη + χσµχ]〉x3(j)

= lima→0

r

2a〈[iηj(χj+1 + χj−1 − 2χj) + iχj(ηj+1 + ηj−1 − 2ηj) + h.c.]〉. (2.17)

The right-hand side is nothing but the “axial symmetry breaker DA0 ” of KS.7 To obtain this

result, note that M = r/a (in the notation of KS, eqn. (5.7) of [13]) since we only have a

Wilson term in one dimension and the bare mass is zero. In addition, we make use of the

identity −ψjγ5ψj′ = ηjχj′ − χj ηj′ for the Dirac spinor (2.10) taken at different points j, j′

along the latticized dimension.

Having thus established the equivalence of the deconstructed theory action to the Wilson

fermion action and having matched the corresponding anomalous Ward identities, we are

assured that the remainder of the weak coupling axial anomaly calculation goes through

according to KS. The latticization of one dimension regulates the linear divergence usually

associated with the weak coupling anomaly; furthermore the regulator respects the vector

current Ward identity and Bose symmetry; hence we are assured that the axial anomaly

emerges just as if we had latticized all 4d, as KS have done.

One difference with the work of KS is worth pointing out. In a nonsupersymmetric lattice

theory, one has the freedom to vary the Wilson term coefficient r at will. For r = 0, the

6Of course, δLfreeferm./δϕj of eqn. (2.4) is obtained from this by taking r = 1.

7Up to an overall normalization. Also note that in the notation of KS, λ0 = 1 corresponds to the

flavor-diagonal current.

9

doublers have a continuum 4d interpretation as massless fermions of opposite axial charge

which cancel the anomaly of the physical fermions. For 0 < r < 1/√2, the doublers still have

a 4d continuum interpretation as fermions of opposite axial charge, but with chiral symmetry

violating masses. This follows from the Wilson fermion dispersion relation, which can be

read off eqn. (2.11), or, more easily, from the geometric D-brane picture, see eqns. (3.27),

(3.28):

a2ω2(k) = sin2 2πk

N+ 4r2 sin4 πk

N. (2.18)

(Here ω is the energy and for simplicity we have set the continuum momenta p1, p2 to zero.)

Thus (for 0 < r < 1/√2) the effect of the doublers can be “physically” described as that of

Pauli-Villars regulators, whose axial symmetry violating mass terms give rise to the anomaly.

In our case, r = 1, it is easy to see that there is no continuum interpretation of the would-be

“doublers” as 4d massive Dirac fermions; nevertheless, as KS showed, the calculation of the

anomaly is valid for all values of r.

3 Fractional Branes and the Chiral Anomaly via Clas-

sical Supergravity

In this section, we show that the lattice perturbation theory result for the U(1)R

anomaly can be obtained via a classical supergravity calculation in the fractional brane

realization of the lattice supersymmetric models. The brane construction is very general and

underlies all proposed realizations of lattice supersymmetry by deconstruction.

For completeness, we begin in section 3.1 by reviewing the fractional brane construction

of the continuum limit 4d N = 2 theory and describing the calculation of the continuum

theory U(1)R anomaly. This section also reviews well-known results about the closed string

calculation of the running coupling in supersymmetric gauge theories.

The fractional brane construction of the deconstructed SU(k)N theory is given in section

3.2. Then, in section 3.3, we study the running of the gauge coupling in the deconstructed

theory. We compare the running coupling calculation (section 3.3.1) with the field theory

(section 3.3.2) and study the continuum limit in the fractional brane picture. The super-

gravity calculation of the anomaly is discussed in section 3.3.3. Finally, in section 3.4, we

give a geometric picture of the R-symmetry and the symmetry-breaking Wilson term.

10

3.1 Anomalies in Seiberg-Witten Theory via Fractional Branes

First, let us derive the non-abelian anomaly and β-function of the Seiberg-Witten theory

from an appropriate orbifold. This will also be the desired result in the continuum limit of

the deconstruction setup.

In order to realize a four dimensional theory with eight supercharges, we take D3 branes

stretched along X0, .., X3. The D3 branes are transverse to a C2/Zr orbifold acting in the

standard way on the coordinates X4, ..., X7. Here r ≥ 3, but otherwise arbitrary.

This gives a quiver gauge theory [4] on the brane, whose matter content depends on the

choice of representation of the Chan-Paton (CP) factors. The simplest such choice, namely

the one-dimensional trivial representation of Zr, gives a pure gauge theory.8 Note that the

more common choice—the regular representation—is r-dimensional, allowing for branes to

move away from the fixed point. Here the brane is stuck at the fixed point (i.e. its world-

volume theory has no Higgs branch), and is dubbed a “fractional brane.” Of course one can

have any multiplicity for these representations, giving non-abelian theories.

These fractional branes behave slightly differently from the usual D3 branes [16]. For

example, when calculating their kinetic term, it is a different modulus than the complexified

string coupling that controls their gauge coupling. This can be demonstrated by an explicit

worldsheet calculation, as sketched below.

The closed string sector has r−1 twisted sectors, and each one of them has in the spectrum

a massless complex scalar (propagating in the 6 dimensions transverse to the orbifold). We

denote these moduli by τi, where for a general orbifold i runs over all the nontrivial elements

of the discrete group. Here i = 0, ..., r − 1, where we denote by τ0 the usual (untwisted)

complexified string coupling.

Generally, to specify a particular quiver theory one chooses a representation of the CP

factors of some dimension, say m. Here m = 1, while for the regular representation m = r

(we discuss here the general case, which will be useful later). This means that every element

of the group is represented by a matrix γi of dimension m. The gauge fields carry a (diagonal)

m × m matrix A. For non-abelian theories we simply take some multiplicity of the above

structure.

One can now proceed to calculate the gauge theory action from various worldsheet cou-

plings. The actual worldsheet amplitude will be only a (non-zero) irrelevant factor. All that

matters is various selection rules one gets from the CP part of the amplitudes.

8Obtaining a theory with matter of the quiver form is done simply by including more representations of

Zr. This should not present any essential new features for our purposes here.

11

For example, in calculating kinetic terms in the action, one has to calculate the amplitude

with two open string vertex operators and one closed string modulus. The dependence on

the CP factors is encoded in the trace Tr(AγiA). For example, the usual case—with the

regular (traceless) representation for γi—has A proportional to the unit matrix, therefore

Tr(AγiA) = 0 unless i = 0. One then deduces that τ0 is the gauge coupling on the brane.

For the fractional brane, on the other hand, the dimension m = 1 and γi = ωir, where

ωr is any r-th root of unity. If we choose for simplicity this root of unity to be 1, then

all the generators of Zr are simply represented by that number. With this choice, our

fractional brane couples identically to all fields τi (i = 0, . . . , r−1). This is more conveniently

summarized if we perform a discrete Fourier transform on the closed string fields:

τm =r−1∑

i=0

ωimr τi . (3.1)

We see therefore that our fractional brane couples to the field τm=0 only. The other possible

choices of r-th root of unity lead to fractional branes, each of which couples to a specific

Fourier mode of the closed string fields.

Having identified the holomorphic gauge coupling, we are now ready to discuss quantum

effects. Suppose one wants to calculate the running coupling of a non-abelian gauge theory

SU(k) at some scale |A|. This can be achieved, for example, by going to a particular

point of its Coulomb branch, where a VEV of magnitude |A| breaks the gauge symmetry to

SU(k − 1)× U(1). The low-energy effective action (valid at scales below |A|) then contains

a kinetic term for the abelian field:∫

d2θ τ(A)W αWα, (3.2)

and the value of the U(1) (frozen) gauge coupling is equal to the SU(k) (running) gauge

coupling, evaluated at the scale of the breaking |A|.In the field theory, the abelian gauge coupling at A (and below) is obtained by integrating

out the massive W-boson supermultiplets which are charged under the U(1). In a brane

realization the SU(k) symmetry lives on a set of k D-branes, which we presently separate to

one group of k − 1 branes, which we call the “source branes,” and a single brane, called the

“probe brane.” The massive W-bosons are then strings stretched between the source and

the probe (s − p strings). We emphasize that the distinction between the source and the

probe is a matter of convenience, and applies even in the k = 2 case.

At one loop, the result can be extracted from an annulus diagram with two massless open-

string vertex operators, corresponding to U(1) gauge bosons, inserted on the boundary. In

12

order to isolate the contributions of the s − p strings alone, the boundary conditions on

both boundaries are chosen differently: one of them (the one with the gauge vertex operator

insertions) in correspondence to the probe brane, and the other in correspondence to the

source branes. The mass of the W-bosons is proportional to the separation between the

branes, so in order to decouple the open-string oscillator modes, one works in the limit of

sub-stringy separations.

The supergravity analysis relies on the old observation that the annulus diagram can

be calculated in the closed-string channel as well. The limit appropriate for a supergravity

treatment is that of large separations compared to the string length, but supersymmetry [17]

aids in making the extrapolation to the sub-stringy, field theory, regime.

In the closed-string channel, the annulus factorizes into a sum of all possible intermediate

closed string fields. In the supergravity limit only the massless fields are relevant. Each

contribution is then a product of three factors:

1. The tadpole: a disc diagram with one closed-string insertion, and boundary conditions

corresponding to the source branes. In the supergravity action this is summarized by a

source term for the closed-string field. From the above discussion we see that only the

field τm=0 has a tadpole. The tadpole in our case is proportional to k, the total number

of branes.9 We choose to normalize the closed-string fields such that the tadpole isb02π

= kπ, where b0 = 2k is the one-loop β function coefficient of SU(k).

2. Closed-string propagator: since the twisted sector fields are allowed to vary only in two

directions, this gives a logarithmic dependence on the coordinates.10 Using complex

notation for the X8, X9 plane, z ≡ X8 + iX9, one gets:

Φ(z) = c+b02π

log(|z|) , (3.3)

where c is an integration constant. Φ(z) is a NS sector scalar, which we relate to the

coupling constant below.

3. The response: a disc diagram with two open-string vertex operators, and one closed-

string vertex operator. This summarizes the coupling of the probe to the closed-string

field. The interaction of the probe occurs at the location z = A, and its strength can be

9And not to k − 1. This comes from the contribution of CP factors to the factorized closed-channel

amplitude: the source’s coupling to the tadpole has a factor of Tr 1 = k.10In our conventions for the normalization of the closed-string field kinetic terms, the (2 dimensional)

propagator satisfies ∇2G2(x, y) = 2πδ(x)δ(y).

13

chosen to be 14π

(by normalizing the open string gauge fields appropriately), therefore

the coupling on the probe is:

1

g2eff=

c

4π+

b08π2

log(|A|) . (3.4)

This indicates that the integration constant c should be related to the bare coupling

by c4π

= 1g20.

We can use holomorphy to deduce the imaginary part of the coupling (3.4). Defining as

usual τ(A) = Θ(A)2π

+ i 8π2

g2(A), one gets,

τ(A) = τ0(A) + i b0 logA. (3.5)

In the fractional brane realization of the N = 2 theory, the anomalous R-symmetry is the

SO(2) rotation in X8, X9 directions, i.e. z → eiαz implies A → eiαA. (The SU(2)R comes

from action on the orbifold directions, or more generally on the complex structures of K3.)

Therefore the result (3.4) encodes both the running coupling and the R-symmetry violation.

3.2 The Deconstruction Setup with Fractional Branes

We now consider the deconstruction of the same theory. Generally, there are two ways to

view deconstruction as a string theory setup:

1. As an orbifold of a Hanany-Witten setup [18], where D-branes are stretched between

NS fivebranes. In the continuum limit (moving asymptotically on the Higgs branch)

the D-branes combine and move away from the orbifold in one direction along the NS

fivebranes. We will not need this setup below, so we will not provide any details here.

2. As a setup of fractional branes on an orbifold. This is the description we adopt as it

allows for a perturbative string theory treatment [19].

In our case—the deconstruction of 4d Seiberg-Witten theory—we start with D2 branes11

stretched along the X0, X1, X2 directions. The orbifold group is chosen to be Zr × ZN (we

take r,N to be relatively prime), which acts as follows:

11The spatial lattice constructions of [6] can be obtained by similar (fractional) brane constructions: the

3d N = 4 theory is obtained upon replacing our fractional D2 branes with fractional D0 branes, and the

C2/Zr × ZN orbifold by a C3/Zr × ZN1× ZN2

orbifold; the 4d N = 4 theory, on the other hand, involves

only regular D0 branes on a C3/ZN1× ZN2

× ZN3orbifold.

14

• The generator of ZN acts on the directions X3 + iX8, X4 + iX5 in the standard way.

In the deconstruction setup one needs to move into the Higgs branch, so our branes

will need to be in the regular representation of ZN .

• The generator of Zr acts on the directions X4+iX5, X6+iX7. The CP assignments for

this generator are as before (section 3.1) only taken with multiplicity of kN , to account

for all images with respect to the ZN orbifold action, and to generate a nonabelian

SU(k) gauge group in four dimensions.

Thus, our CP matrices are kN dimensional. Let us work out the CP factors corresponding

to the various bosonic fields. Denote the action of the generator ZN on the CP factors by

γN and take the regular representation, tr γN = 0:

γN = diag(ωN , ω2N , . . . , ω

NN ) , (3.6)

where every element is multiplied by a k×k unit matrix. The gauge field’s (Am, m = 0, 1, 2)

CP factors are (generally complex) kN × kN matrices, which we denote by A. They obey:

A = γNAγ−1N , (3.7)

which means that the states that are not projected out by the ZN orbifold have block diagonal

CP factors:

A =

λ1 0 . . . 0 0

0 λ2 . . . 0 0

0 0 ...λi... 0 0

0 0 . . . λN−1 0

0 0 . . . 0 λN

, (3.8)

where the λi are the k × k CP matrices of the unbroken U(k)N gauge groups.

Similarly, we take as the generator of the Zr action γr = diag(ωr, . . . , ωr), with ωr the

one-dimensional representation of Zr. It is evident that all gauge boson states with CP

factors (3.8) survive the Zr orbifold projection.

We now move on to describing the scalar fields. One scalar field that survives the com-

bined ZN × Zr action corresponds to motion of the D2 branes in X9 (the scalar of the 3d

N = 2 vector multiplet). This scalar is described by the same CP factor as the gauge field

(3.8).

15

The other set of surviving scalars correspond to moving in the X3 + iX8 directions, and

make up the link fields (bifundamentals) of the deconstruction setup. The link fields CP

factors Φ obey:

Φ = ωNγN Φ γ−1N , (3.9)

which is solved by:

Φ =

0 φ1,2 0 . . . 0 0

0 0 φ2,3 . . . 0 0

0 0 0 ...φi,i+1... 0 0

0 0 0 . . . 0 φN−1,N

φN,1 0 0 . . . 0 0

, (3.10)

where φi,i+1 are the k × k CP matrices of the N link fields. In fact, the CP factors of

the link fields above correspond to our fields φi,i+1 → Qi of eqn. (2.1) in the coordinate

representation.

The moduli space of the kN D2 branes on the orbifold is three-dimensional—branes are

allowed to move in groups of k, in a ZN symmetric way in X3 + iX8 (Higgs branch), as

well as in X9 (Coulomb branch). The deconstructed theory is obtained upon moving them

a distance v from the origin in X3+ iX8, i.e. along the Higgs branch, as explicitly described

in section 2.1.

To proceed with the analysis of gauge coupling running and the anomaly, we need the

couplings of the world-volume open string massless modes to the closed string fields. This is

a particular example of compactification of type IIA string theory on a Calabi-Yau manifold,

and the closed string spectrum is well-known (see e.g. [4]); we review the orbifold analysis

for the sake of completeness.

A priori there are Nr twisted sectors of closed strings to consider. However, to be

sourced by our D-brane setup, the closed string fields need to be a particular combination

of the twisted sectors with respect to the Zr orbifold, as described in the previous section.

Therefore, for our purposes there are only N non-trivial sectors to consider. These are

N − 1 twisted sectors, and one untwisted sector, with respect to the ZN action. To see the

matter content in these sectors it is useful to ignore the ZN orbifold at first. In this case

the twisted sector fields fill out six dimensional (1, 1) vector multiplets.12 The bosonic fields

in each one of those multiplets consist of a real vector (from the R-R sector) and 4 scalars

(from the NS-NS sector).12The untwisted sector contains also a 6 dimensional gravity multiplet, which decouples from our analysis.

16

Subsequently imposing the ZN projection yields a 3-dimensional N = 4 vector multiplet

for each of the N sectors relevant for us.13 The bosonic closed string fields surviving the

complete orbifold projection are then complex scalar fields Ψi (from the NS-NS sector), a

vector field Vi (from the R-R sector) and a real 0-form Si (from the R-R sector), where

i = 0, ..., N − 1. As usual, the i = 0 case refers to the untwisted sector with respect to ZN ,

and as mentioned above all the fields are assumed to be the correct linear combination of

twisted sectors with respect to the Zr action (the m = 0 Fourier mode, in the notations of

the previous section, eqn. (3.26)).

The charges of the D-branes with respect to these fields can be calculated by a disc

diagram with no external states and a closed string twisted state vertex operator insertion.

Since we are interested in the configuration of separated fractional branes, we calculate the

charges of each fractional brane separately.

The only new element here is the factor that comes from the CP matrices of the ZN

action. If we are interested in the l-th fractional brane, then the generator of the ZN action

is represented by the CP factor wlN 1k×k. Therefore the coupling to any of the q-th twisted

sector (with respect to ZN) closed string fields is proportional14 to kNwql

N .

The four dimensional couplings, in the continuum limit, are T-dual to the closed string

fields in our brane setup. It is clear then that we are interested only in the untwisted sector

fields q = 0 (with respect to ZN).15 We denote these fields by Ψ (a complex scalar field), V

(a vector field), and S (a 0-form). We choose the normalizations of the closed string fields

such that the coupling of the individual fractional branes to these fields is b0N, with no phase

factors. This follows from the standard relation between different brane tensions (T2 = 2πT3,

for α′ = 1).

3.3 The Running Coupling and the Anomaly

3.3.1 Supergravity Calculation

Let us elaborate on the equations of motion for the closed string fields sourced by the D2

brane configuration (shown in Figure 1). The sources for the tadpole are localized in the

13The untwisted sector with respect to ZN contains an additional hypermultiplet, which is irrelevant for

our analysis.14Note that we choose the trace over the CP matrices to be normalized such that Tr(1N×N) = 1. In

particular, in these conventions the charge of a brane in the regular representation (which has all N images)

is chosen to be 1.15These fields are the parameters of the 3 dimensional action.

17

X3, X8, X9 space at the locations:

X3k = v sin

2πk

N,

X8k = v cos

2πk

N, (3.11)

X9k = 0 ,

i.e. the branes are distributed with equal spacing on a circle of circumference 2πv in the

X3–X8 plane. Each one of the source terms is of strength b0N

in the normalization of the last

X

k=0

k=1

k=2

k=N−1

v

2π/Ν

3

X8

Figure 1: D-brane sources of the Zr-twisted fields Ψ, (3.12), and U , (3.23), in a ZN -symmetric

distribution along a circle of radius v in the X3-X8 plane, at X9 = 0. The Coulomb branch

direction X9 is perpendicular to the drawing. The branes are labeled by values of k, as defined in

(3.11).

section. The NS-NS sector scalar field Ψ couples to the branes as electric sources,16 i.e.:

∇2Ψ =b0N

N−1∑

k=0

δ(3)(X −Xk) , (3.12)

16We choose to normalize coordinates such that the 3d propagator satisfies∇2G3 = δ(x)δ(y)δ(z); therefore

we have in comparison to the previous sectionR∫

−R

dz G3(x, y, z) ≃ 2πG2(xR, y

R) for x, y ≪ R.

18

which yields for the value of Ψ at arbitrary X3, X8, X9:

Ψ(X3, X8, X9) = C − b04πN

N−1∑

k=0

1√

(X3 −X3k)

2 + (X8 −X8k)

2 + (X9)2. (3.13)

where C is a constant.

In order to see a 4d behavior, we need to focus to a neighborhood close enough to the

circle of D-branes, such that in the large-N , large-v limit, a nearby section of the circle

approximates a straight line. To that effect, we place the probe brane at a position close to

the sources. Since the probe brane is also in the regular representation with respect to ZN ,

this will include all images of that brane

The continuum limit of the deconstructed theory is best exhibited upon introducing

appropriate, see Figure 2, coordinates17 (everything here and below is made dimensionless

in terms of the string length ls) ρ, η, φ defined as:

X3 = (v + ρ) sin φ ,

X8 = (v + ρ) cos φ , (3.14)

X9 = η .

The probe brane is placed18 at constant ρ, η, and at φ = 2πk′

N, with k′ = 0, . . . , N−1 counting

its images.

In contrast to the previous section, the normalization of 3-dimensional gauge fields is

chosen such that the probe couples to Ψ with strength 12. In particular, with these conventions

the 3 dimensional actions does not have an overall 1N

factor.

In the absence of the source, the gauge coupling on the probe brane is determined by the

constant of integration, C of eqn. (3.13). With the source, one finds for the gauge coupling

on the probe:

1

g2probe=

1

2

N∑

k′=1

Ψ(ρ, η, φ =2πk′

N)

=NC

2− b0

16πv

N−12∑

(k−k′)=−N−12

[

(

1 +ρ

v

)

sin2

(

π(k − k′)

N

)

+ρ2 + η2

4v2

]− 12

(3.15)

≃ NC

2− b0N

16π2v

π2∫

−π2

dx√

sin2 x+ ρ2+η2

4v2

≃ NC

2+b0N

8π2vlog

|ρ+ iη|2v

,

17See also discussion about coordinate choice in section 3.418This choice of φ-coordinate for the position of the probe brane indicates that we do not turn on a Wilson

line for the 4d gauge field even at finite R, since, in any case, it would vanish in the continuum R → ∞ limit.

19

X

φ

ρv

X

X = v, X

= 0

3

88

9

Figure 2: The location of a probe brane (at general φ), shown here by an open circle, in relation

to the arc of source branes in the neighborhood of X8 = v, X3 = X9 = 0. According to (3.14), ρ

measures the radial displacement of the probe brane from this arc (of radius v), while φ gives the

angular displacement from the X8 axis. The X9 displacement η of the probe brane is not shown;

the figure should be thought of as a projection onto the X9 = 0 plane. The vertical line passing

through X8 = v is the R-symmetry axis of section 3.4.

where we took 1+ρ/v ≃ 1, and replaced the sum19 by an integral in the large-N limit. Note

that the last equality in (3.15) is only valid for |ρ+ iη| ≪ v; the tadpole falls off as an inverse

power of |ρ+ iη| at large values of the argument.

In the next section, we will see that the probe-brane coupling, eqn. (3.15), reproduces

the running coupling of the continuum theory and will relate the constant C to the “bare”

4d coupling. In order to continue with the interpretation of eqn. (3.15), we thus remind

ourselves of the deconstructed field theory calculation of the running coupling.

3.3.2 Field Theory Calculation and the Continuum Limit

We want to relate the tadpole Ψ(ρ, η, φ) of eqn. (3.15) to the running of the gauge coupling

in the field theory, evaluated at the scale |A| = |ρ+ iη|. The gauge coupling on a probe brane

placed at ρ + iη, away from the circle, will depend on the background Ψ, as described in

19See [20].

20

section 3.2 for the continuum limit theory. For simplicity we will only focus on the SU(2)N

case.

The probe brane gauge coupling is the coupling of the unbroken U(1) on the Coulomb

branch of the diagonal SU(2) ⊂ SU(2)N 3d gauge group—moving a brane away from the

circle of radius v in the X3, X8 plane is equivalent to going on the Coulomb branch. The

running of the coupling is due to the heavy gauge bosons and superpartners which acquire

masses on the Coulomb branch. The holomorphic gauge coupling “runs” only at one loop—

more precisely, upon integrating out a (3d N = 4) vector multiplet of mass m, the 3d

diagonal gauge coupling g3,D receives a threshold correction:

1

g23,D low

=1

g23,D high

− 1

2πm, (3.16)

where the coefficient was calculated in [21]; the minus sign indicates that the 3d theory

becomes more strongly coupled in the infrared.

To calculate the gauge coupling of the U(1) at the scale A, we have to sum the thresholds

(3.16) from all heavy states charged under the U(1), the W± bosons. We start with the tree

level value of the diagonal gauge coupling at the cutoff scale20 v:

1

g23,D=N

g23, (3.17)

as is usual in deconstruction. A short calculation gives for the W±k boson masses along the

Coulomb branch, for |A| ≪ v:

(m±k )

2 = 4v2 sin2 πk

N+ (ρ2 + η2) = 4v2 sin2 πk

N+ |A|2. (3.18)

The coupling of the unbroken diagonal U(1) at the scale |A| = |ρ+ iη| is given by integrating

out all W± since they are all heavier than A; recall that we are working in a normalization

where 1/g23 multiplies the entire lagrangian (2.1, 2.2):

1

g23,D(|A|)=

N

g23− 1

2π

N−12∑

k=−N−12

1√

4v2 sin2 πkN

+ |A|2

≃ N

g23− N

4π2v

π2∫

−π2

dx√

sin2 x+ |A|2

4v2

. (3.19)

20Hereafter, our normalization of the vev (2.3) of Qi differs from that of section 2: in the brane picture,

it is conventional to have 2πR = N/v, as opposed to (2.9).

21

This is, of course, a regulated version of the continuum calculation, where (in the first

equality, before replacing the sum with an integral) we would have an infinite sum of Kaluza-

Klein tower thresholds (3.16) with mk = k2R−2 + |A|2.The diagonal gauge coupling g23,D(|A|) of the deconstructed theory (3.19) is identified,

upon dimensional reduction, with the running coupling of the continuum theory, see [14],

[10]. The 4d running coupling 1/g24(|A|) is thus obtained upon multiplication of (3.19) by

(2πR)−1 and replacing the sum with an integral at large N :

1

g24(|A|)≡ v

Ng23,D(|A|)=

v

g23− 1

4π2

π2∫

−π2

dx√

sin2 x+ |A|2

4v2

=v

g23+

1

2π2ln

|A|2v

. (3.20)

The coefficient in front of the logarithm is the one appropriate for the 4d N = 2 pure SU(2)

Yang-Mills theory, which is usually written as b0/(8π2), with b0 = 4. The continuum limit

in the deconstructed theory is then achieved by taking (with Λ fixed):

N → ∞, v → ∞,g23v

→ 0, Λ4 = v4 e− 8π2v

g23 , Λ ≪ v . (3.21)

Finally, to compare with the supergravity calculation, we note that the coupling on the

probe brane, eqn. (3.15) with b0 = 4, is identical to the coupling of the diagonal 3d gauge

group (3.19). Comparing (3.19) with the tadpole (3.15) also gives a precise relation between

the integration constant C and the tree level coupling of the 3d theory: C/2 = 1/g23.

3.3.3 Supergravity and the U(1)R Anomaly

It is clear from the supersymmetry of the problem that supergravity should also yield the

correct expression for the U(1)R anomaly in the continuum limit, similar to the N = 2 case

discussed in section 3.1. Here, we briefly elaborate on this.

It is most convenient for the following discussion to work in terms of the six dimensional

theory one gets by orbifolding with respect to Zr only. The closed string spectrum contains

a twisted R-R 1-form, which we denote U .21 The fractional D2 branes are magnetic sources

for this field (or electric sources for its dual C(3)). The D2 branes are defined as having:

∫

S2dU =

b0N

(3.22)

21This field is untwisted with respect to ZN , and therefore propagates in 6 dimensions. It is a combination

of fields from the 3 dimensional hyper- and vector multiplets. Like all closed string fields relevant for us, it

is in the m = 0 Fourier mode with respect to the Zr action.

22

where S2 is any two sphere in the 3, 8, 9 space surrounding only one 2-brane. The factor b0N

is the R-R charge of a fractional brane, as computed in section 3.2.

The tadpole of the R-R twisted vector is determined by the N sources at coordinates Xk

(3.11), similar to (3.12):

(dU)ab ∼ b0N

ǫabc ∂cN∑

k=1

1

|X−Xk|, (3.23)

in an obvious notation. Clearly, the field U is the sum of the vector potentials for magnetic

monopoles located at Xk. We will not need the general expression for U , since near the probe

brane and in the large-N limit the r.h.s. of eqn. (3.23) simplifies: as we already showed in

our discussion of the scalar tadpole (3.15), there∑N

k=11

|Xa−Xak|∼ log |ρ+ iη|, where ρ and η

are essentially X8 and X9 parametrizing the tangent plane to the circle. Indexing ρ, η by

i, j = 1, 2, we have from (3.23):

(dU)i3 ∼ ǫij3 ∂j log |ρ+ iη| . (3.24)

Thus, near the probe brane the expression for the RR twisted tadpole simplifies:

U3 ∼ arg(ρ+ iη) . (3.25)

The Chern-Simons coupling to the twisted RR field on the probe brane world-volume∫

V3

U∧F[4] leads, in the classical background (3.25), to:

∫

d3x U3 ∂mX3 Fpq ǫ

mpq ∼ argA∫

d3x ∂mX3 Fpq ǫ

mpq (3.26)

and gives rise to the 4d anomaly (recall that X3, the position of the brane in the compact

direction, becomes the third component of the gauge field) exactly as in the N = 2 case

discussed in section 2.1.

3.4 R-symmetry and the Wilson Term: a Geometrical Picture

Issues pertaining to fermion doublers and the anomalous R-symmetry away from the con-

tinuum limit are best visualized in the string theory embedding described above. Though

it is simple enough to resolve such issues in the present context purely by field theoretical

means, the stringy methods are expected to help in more complex situations.

In our setup we have N fractional branes which are distributed in an array in the X3, X8

plane, which is in turn embedded in the X3, X8, X9 space. In the continuum limit the branes

23

are distributed evenly on an approximate (for large N) circle, which is to be considered as

the direction conjugate (or T-dual) to the compact deconstructed direction.

Let us consider the open string fields on the fractional branes. On each brane one has

3 massless scalar fields, corresponding to fluctuations of the brane in the space spanned

by X3, X8, X9. In the continuum limit two of those are to be considered as a complex 4-

dimensional scalar field, and the remaining one as a Wilson line of the 4-dimensional gauge

field around the compact direction.

A priori this is a local freedom, at each location of a fractional brane. This is represented

in the field theory language as a field redefinition freedom, which was used extensively in

section 2. Such choices differ, away from the continuum limit, by the value of the Wilson

line turned on.

We choose to identify the complex scalar field as the fluctuations in the X8, X9 direction,

away from the location of a preferred brane, say the one with k = 0. In other words we

identify ρ, η, defined above, as the complex scalar field. This corresponds to turning off the

Wilson line, and as we see shortly, reproduces the Wilson R-symmetry breaking term.

The identification of open string fields corresponds to a definition of the R-symmetry

transformation away from the continuum limit. In general an R-symmetry transformation

will be a rotation around some axis in the X3, X8, X9 space. Our choice of fields corresponds

to choosing the axis of rotation to be tangent to the circle of branes at the point k = 0.

It is clear then, that the locations of the k 6= 0 branes away from the chosen tangent

to the circle breaks the R-symmetry. Consider for definiteness the tangent at the position

of the k = 0 brane, which, in the coordinates of (3.11), is at X3 = 0, X8 = v. The

continuum limit R-symmetry is then an SO(2) rotation in the X3, X8, X9 space around the

axis X8 = v,X9 = 0 (this axis is shown on Figure 2).

Recall now that the locations of the branes in the X3–X8 plane correspond to mass

terms in the deconstructed theory. Having each momentum mode arise from a brane at a

different location in X8 corresponds exactly to the (supersymmetric completion of) Wilson

R-symmetry-breaking term. The masses of the adjoint supermultiplet due to strings between

the 0-th and k-th brane are equal to the lengths of the strings stretched between the two

branes. The fermion mass itself is complex and can be written as:

mk = ei2πkN v − v = (X8

k − v) + i X3k , (3.27)

where X3(8)k are the locations (3.11) of the k-th brane. The utility of writing the mass this

way is to separate the R-symmetry preserving (parallel to X3) and R-symmetry breaking

24

(perpendicular to X3) parts. The dispersion relation that follows from (3.27) is:

v−2ω2(k) = v−2 |mk|2 = sin2 2πk

N+ 4 sin4 πk

N. (3.28)

Comparing (3.28) with (2.18) identifies the Wilson term coefficient r = 1. The value of the

Wilson term coefficient is thus of order the inverse lattice spacing, is fixed by supersymmetry

and is not a free parameter.

Acknowledgements

This work is supported by the National Science and Engineering Research Council of Canada.

E.P. acknowledges the hospitality and support of the “Braneworld and Supersymmetry”

Workshop at the Pacific Institute for Mathematical Sciences during the initial stages of this

work.

References

[1] N. Arkani-Hamed, A. G. Cohen and H. Georgi, “(De)constructing dimensions,” Phys.

Rev. Lett. 86, 4757 (2001) [arXiv:hep-th/0104005];

C. T. Hill, S. Pokorski and J. Wang, “Gauge invariant effective Lagrangian for Kaluza-

Klein modes,” Phys. Rev. D 64, 105005 (2001) [arXiv:hep-th/0104035].

[2] H. Georgi and A. Pais, “CP - Violation As A Quantum Effect,” Phys. Rev. D 10, 1246

(1974).

[3] M. B. Halpern and W. Siegel, “Electromagnetism As A Strong Interaction,” Phys. Rev.

D 11, 2967 (1975).

[4] M. R. Douglas and G. W. Moore, “D-branes, Quivers, and ALE Instantons,”

arXiv:hep-th/9603167.

[5] N. Arkani-Hamed, A. G. Cohen, D. B. Kaplan, A. Karch and L. Motl, “Deconstructing

(2,0) and little string theories,” arXiv:hep-th/0110146.

[6] D. B. Kaplan, E. Katz and M. Unsal, “Supersymmetry on a spatial lattice,”

arXiv:hep-lat/0206019.

25

[7] A. Cohen, private communication.

[8] I. Rothstein and W. Skiba, “Mother moose: Generating extra dimensions from simple

groups at large N,” Phys. Rev. D 65, 065002 (2002) [arXiv:hep-th/0109175].

[9] A. Feo, “Supersymmetry on the lattice,” arXiv:hep-lat/0210015.

[10] C. Csaki, J. Erlich, V. V. Khoze, E. Poppitz, Y. Shadmi and Y. Shirman, “Exact

results in 5D from instantons and deconstruction,” Phys. Rev. D 65, 085033 (2002)

[arXiv:hep-th/0110188].

[11] W. Skiba and D. Smith, “Localized fermions and anomaly inflow via deconstruction,”

Phys. Rev. D 65, 095002 (2002) [arXiv:hep-ph/0201056].

[12] N. R. Constable, J. Erdmenger, Z. Guralnik and I. Kirsch, “(De)constructing intersect-

ing M5-branes,” arXiv:hep-th/0212136.

[13] L. H. Karsten, J. Smit, “Lattice fermions: species doubling, chiral invariance and the

triangle anomaly,” Nucl. Phys. B 183 (1981) 103.

[14] C. Csaki, J. Erlich, C. Grojean and G. D. Kribs, “4D constructions of supersym-

metric extra dimensions and gaugino mediation,” Phys. Rev. D 65, 015003 (2002)

[arXiv:hep-ph/0106044].

[15] C. T. Hill and A. K. Leibovich, “Deconstructing 5-D QED,” Phys. Rev. D 66, 016006

(2002) [arXiv:hep-ph/0205057].

[16] D. E. Diaconescu, M. R. Douglas and J. Gomis, “Fractional branes and wrapped

branes,” JHEP 9802, 013 (1998) [arXiv:hep-th/9712230].

[17] M. R. Douglas and M. Li, “D-Brane Realization of N=2 Super Yang-Mills Theory in

Four Dimensions,” arXiv:hep-th/9604041.

[18] J. Lykken, E. Poppitz and S. P. Trivedi, “Chiral gauge theories from D-branes,” Phys.

Lett. B 416, 286 (1998) [arXiv:hep-th/9708134].

[19] R. G. Leigh and M. Rozali, “Brane boxes, anomalies, bending and tadpoles,” Phys.

Rev. D 59, 026004 (1999) [arXiv:hep-th/9807082].

[20] K. Sfetsos, “Branes for Higgs phases and exact conformal field theories,” JHEP 9901,

015 (1999) [arXiv:hep-th/9811167].

26

[21] N. Dorey, D. Tong and S. Vandoren, “Instanton effects in three-dimensional supersym-

metric gauge theories with matter,” JHEP 9804, 005 (1998) [arXiv:hep-th/9803065].

27